- Research
- Open Access
Dynamic analysis of unilateral diffusion Gompertz model with impulsive control strategy
- Yaning Li^{1},
- Huidong Cheng^{1}Email author,
- Jianmei Wang^{1} and
- Yanhui Wang^{1, 2}
https://doi.org/10.1186/s13662-018-1484-3
© The Author(s) 2018
- Received: 29 September 2017
- Accepted: 11 January 2018
- Published: 22 January 2018
Abstract
In this paper, we establish a unilateral diffusion Gompertz model of a single population in two patches in a theoretical way. Firstly, we prove the existence and uniqueness of an order-one periodic solution by the geometry theory of differential equations and the method of successor function. Secondly, we prove the stability of the order-one periodic solution by imitating the theory of the limit cycle of an ordinary differential equation. Finally, we verify the theoretical results by numerical simulations.
Keywords
- unilateral diffusion
- order-one periodic solution
- limit cycle
- Gompertz model
MSC
- 34C25
- 34D20
- 92B05
- 34A37
1 Introduction
Modern biologists think that the habitat has been deteriorated by an excessive exploitation of resource, and to dodge predators and find suitable habitat, the migration of population becomes necessary [1]. Many researchers focus on this phenomenon and have done a lot of work on it. Skellam [2] investigated the random-walk issue of biological migration and concluded that the balance of the system is related to the size of the habitat. In 1974, Levin and Paine [3] established models of plaque-migratory population dynamics and discussed the influence of the diffusion of population on its survival and extinction. With the development of the research concerned with population diffusion model, more and more people pay attention to diffusion systems. In 1989, Freedman et al. [4] discussed the diffusion system of a single population between two patches and showed that there exists a continuous global asymptotic stable state. In 1994, Zeng et al. [5] focused on the continuous time-diffusion systems and obtained that such a system has a positive and periodic solution.
However, in recent years people have found that the continuous time-diffusion model is not an appropriate description of certain biological phenomena. Some populations spread from one patch to another patch at a fixed time such as the seasonal migration of birds and fish migration. With this in mind, the mathematical model of diffusion with pulse has been established, which makes the research more significant [6–14]. In particular, impulsive differential equations are very important in the research of population migration phenomenon [15–24]. Jiao et al. [25] established a SIR model with pulse vaccination and proved the existence of a disease-free periodic solution, and a large pulse vaccination rate was a sufficient condition to eradicate the disease. Zou et al. [26] considered a population dynamic system with the delays and impulses and obtained sufficient conditions for the coexistence of two populations. For more applications of differential equations, see [27–38].
The paper is organized as follows. In Section 2, we recall some basic results about order-one periodic solutions. In Section 3, we discuss the existence and uniqueness of an order-one periodic solution of system (1) in terms of geometry theory of differential equations and monotonicity of successor function. The stability of an order-one periodic solution is discussed by imitating the theory of the limit cycle of an ordinary differential equation in Section 4. In Section 5, we verify the theoretical results by numerical simulation an make a conclusion.
2 Preliminaries
Definition 2.1
([43])
Remark 2.1
For system (1), we have \(M_{I}\{x,y\}=\{ (x,y)\in R_{2}^{+}|x=\tau,y\geq0\}\) and \(M_{I}^{+}\{x,y\} = \{(x,y)\in R_{2}^{+}|x=(1-c)\tau,y\geq0\}\). The impulse mapping is \(\varphi: (x,y)\in M_{I}\rightarrow({(1-c)\tau}, {c\tau+y_{A}})\in R_{2}^{+}\).
Definition 2.2
([43])
According to Lemma 2.1, we obtain the following conclusions.
Definition 2.3
([43])
If there exists a point B of system (1), then we say that the trajectory passing through point B is an order-one periodic solution of system (1) if \(f(B)=0\).
Lemma 2.2
([43])
The system has an order-one periodic solution if there exist two points \({C}\in{M_{I}^{+}}\) and \({D}\in{M_{I}^{+}}\) such that the successor function \({f({C})f({D})}<0\).
Remark 2.2
This lemma is used to prove the existence of order-one periodic solution of system (1) in Section 3.
Lemma 2.3
([43])
If the successor function of system (1) is a monotonic function, then there exists a unique order-one periodic solution.
Remark 2.3
In Section 3, this lemma is used to prove the uniqueness of order-one periodic solution of system (1).
Proof
In this paper, we restrict ourselves to the biologically meaningful region \(F=\{(x,y)| {x\geq0},{y\geq0}\}\).
3 Existence and uniqueness of order-one periodic solution of system (1)
Lemma 3.1
If \(h<{\frac{a\tau\ln (\frac{K}{x} )}{b}}\), then \(k_{{E^{\prime}}{E^{+}}}< k_{E^{\prime}}\).
Proof
Remark 3.1
The point \(E^{+}\) is under the point E when the slope of the trajectory of system (1) at the point \(E'\) is greater than the slope of the line \(E'E^{+}\), and so we prove that the successor function of point E is negative.
Lemma 3.2
If \(h>\frac{aK}{ec}\), then the straight line \(E'E^{+}\) is not a cutting line of system (1).
Proof
Theorem 3.1
Proof
In summary, according to Lemma 2.2, in the pulse set \(M_{I}^{+}\{x,y\}\), there exists a point B between the points E and M such that \(f(B)=0\), that is, system (1) has an order-one periodic solution. □
Theorem 3.2
System (1) has a unique order-one periodic solution if \(\frac{aK}{ec}< h<\frac{a\tau\ln (\frac{K}{\tau} )}{b}\) and the successor function of system (1) is monotonic.
Proof
Because the successor function of system (1) is monotonic, an order-one periodic solution of system (1) is unique. The proof is completed. □
4 Stability of the order-one periodic solution
Definition 4.1
([42] (Ω limit set))
On the positive half-trajectory of system denoted by \(g(p,T^{+}), T^{+}=(0,+\infty)\), let \(\{0\leq{t_{1}}<{t_{2}}<\cdots<{t_{n}}<\cdots\} \) be a time series such that \(\lim_{n\rightarrow+\infty}t_{n}=+\infty\). If \(P^{*}\) is a limit point of the point range \({g(p,t_{n})}\), \(n=1,2,\ldots\) , then we say that \(P^{*}\) is an Ω limit point. The set of all Ω limit points is called the Ω limit set.
Definition 4.2
([42])
Suppose \(\Gamma'\) is an order-one periodic solution of semicontinuous dynamic system. This periodic solution is called stable if it has a sufficiently small neighborhood \(\bigcup(\Gamma')\) such that the Ω limit set of trajectories starting from any point \(Q\in\bigcup(\Gamma')\) is \(\Gamma'\).
Proposition 4.1
([43])
For system (1), there exists an order-one periodic solution such that its trajectory is through the point \(A'\) in the phase set \(M_{I}^{+}\). For any point \(T_{0}^{*}\) sufficiently close to the point \(A^{\prime}\), there exists a point series \(T_{0}^{*}, T_{1}^{*}, \ldots, T_{k}^{*}, T_{k+1}^{*}, \ldots \rightarrow A^{\prime} (k\rightarrow\infty)\), that is, \(t_{0}, t_{1}, \ldots, t_{k}, t_{k+1}, \ldots \rightarrow 0\), and thus the order-one periodic solution of system (1) is stable. If \(t_{i}<0\) (\(t_{i}>0\)) for \(i=0,1,\ldots ,k,\ldots\) , then the order-one periodic solution is unidirectionally stable.
Using the result in [43] and a similar method as in [42], we can prove the following results, which will prove the main result of Theorem 4.3. The main idea is to set up a rectangular coordinate to debate the theorems for ascertain the stability of order-one periodic solution.
Theorem 4.1
([43])
Theorem 4.2
([43])
Proof
Thus, we get the following conclusions.
Theorem 4.3
([43])
Proof
Therefore the order-one periodic solution of system (1) is stable. The proof is completed. □
5 Numerical simulations and conclusion
5.1 Numerical simulations
In Sections 3 and 4, we have proved that system (1) has a unique stable order-one periodic solution. Here we give an example to verify the results.
5.2 Conclusion
In the previous studies, we always used the Poincaré theorem to prove the stability of order-one periodic solution. However, this method is not suitable for all systems.
In this paper, we compare the slope of pulse line and the trajectory at the pulse point. We get the condition that the successor function is negative, and we prove that system (1) has an order-one periodic solution.
Secondly, we use the monotonicity of the successor function, the geometric theory of differential equations, and the definition of the successor function to prove the uniqueness of an order-one periodic solution.
Finally, we consider a new method to prove the stability of an order-one periodic solution. In Proposition 4.1, the unidirectional stability of an order-one periodic solution can be used to determine that there is no closed orbit around the order-one periodic solution. Using the method of approximating time, we get the same conclusion of the limit cycle theory. So we rewrite Theorem 4.1 to Theorem 4.3. This method solves the problem that the subsequent function is difficult to calculate in the Cartesian coordinate system and also makes the problem simple.
Our main purpose is to prove the existence, uniqueness, and stability of an order-one periodic solution of system (1). However, does there exist a periodic solution for the migration of two- or multipopulation? If it does, then is the periodic solution stable? We leave these for our future work.
Declarations
Acknowledgements
The paper was supported by the National Natural Science Foundation of China (No. 11371230, 11501331), Shandong Provincial Natural Science Foundation, China (No. S2015SF002), SDUST Research Fund (2014TDJH102), and Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province of China.
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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